A logarithmic analogue of Alladi's formula

2025/04/22 by Wang, Biao
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.16002

Abstract

Let μ(n) be the Möbius function. Let P^-(n) denote the smallest prime factor of an integer n. In 1977, Alladi established the following formula related to the prime number theorem for arithmetic progressions -∑_\substackn≥ 2
P^-(n)≡ ℓ (\rm modk)(μ(n))/(n)=\frac1φ(k) for positive integers ℓ, k≥ with (ℓ,k)=1, where φ is Euler's totient function. In this note, we will show a logarithmic analogue of Alladi's formula in an elementary proof.

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